Trigonometry Formulas for Class 10, 11 and 12 — All Identities and Ratios
Trigonometry formulas cover ratios (sin, cos, tan, cosec, sec, cot), standard angle values, and all major identities — Pythagorean, compound angle, double angle, half angle, triple angle, product-to-sum, sum-to-product, and inverse trigonometry. These are essential for CBSE Class 10, 11, 12, JEE Main, JEE Advanced, and NEET.
Topics Covered on This Page:- Trigonometric ratios and standard angle values table (0° to 360°)
- Pythagorean, reciprocal, and quotient identities
- Compound angle and double angle formulas
- Half angle, triple angle, product-to-sum and sum-to-product formulas
- Inverse trigonometry formulas with domain and range (Class 12)
- Class-wise NCERT chapter reference
What are Trigonometry Formulas?
Trigonometry formulas are mathematical relationships between the angles and sides of a right-angled triangle. The six trigonometric ratios — sine, cosine, tangent, cosecant, secant, and cotangent — are defined as ratios of two sides of a right triangle with respect to one of its acute angles. Using these ratios, we derive identities that hold true for all values of the angle. Trigonometry is first introduced in NCERT Class 10 (Chapter 8) and developed further in Class 11 (Chapter 3) and Class 12 (Chapter 2).
Standard Angle Values Table
Memorise this table. These exact values are directly tested in CBSE board exams and are needed to apply all other trigonometry identities quickly.
| Angle | sin | cos | tan | cosec | sec | cot |
|---|
| 0° | 0 | 1 | 0 | — | 1 | — |
| 30° | \( \frac{1}{2} \) | \( \frac{\sqrt{3}}{2} \) | \( \frac{1}{\sqrt{3}} \) | 2 | \( \frac{2}{\sqrt{3}} \) | \( \sqrt{3} \) |
| 45° | \( \frac{1}{\sqrt{2}} \) | \( \frac{1}{\sqrt{2}} \) | 1 | \( \sqrt{2} \) | \( \sqrt{2} \) | 1 |
| 60° | \( \frac{\sqrt{3}}{2} \) | \( \frac{1}{2} \) | \( \sqrt{3} \) | \( \frac{2}{\sqrt{3}} \) | 2 | \( \frac{1}{\sqrt{3}} \) |
| 90° | 1 | 0 | — | 1 | — | 0 |
| 180° | 0 | −1 | 0 | — | −1 | — |
| 270° | −1 | 0 | — | −1 | — | 0 |
| 360° | 0 | 1 | 0 | — | 1 | — |
Pythagorean Identities
These three identities follow directly from the Pythagoras theorem. They are the foundation of all trigonometric proofs and are tested in every CBSE board exam.
| Identity | Formula | Derived From |
|---|
| Sine and Cosine | \( \sin^2\theta + \cos^2\theta = 1 \) | Unit circle definition |
| Tangent and Secant | \( 1 + \tan^2\theta = \sec^2\theta \) | Divide identity 1 by \( \cos^2\theta \) |
| Cotangent and Cosecant | \( 1 + \cot^2\theta = \text{cosec}^2\theta \) | Divide identity 1 by \( \sin^2\theta \) |
Reciprocal and Quotient Identities
| Ratio | Reciprocal Formula | Quotient Formula |
|---|
| sin | \( \sin\theta = \frac{1}{\text{cosec}\,\theta} \) | \( \sin\theta = \frac{\cos\theta}{\cot\theta} \) |
| cos | \( \cos\theta = \frac{1}{\sec\theta} \) | \( \cos\theta = \frac{\sin\theta}{\tan\theta} \) |
| tan | \( \tan\theta = \frac{1}{\cot\theta} \) | \( \tan\theta = \frac{\sin\theta}{\cos\theta} \) |
Compound Angle Formulas
These formulas give the trig ratio of the sum or difference of two angles A and B. They are derived from the unit circle and are directly tested in JEE Main.
| Formula | Expression |
|---|
| \( \sin(A + B) \) | \( \sin A \cos B + \cos A \sin B \) |
| \( \sin(A – B) \) | \( \sin A \cos B – \cos A \sin B \) |
| \( \cos(A + B) \) | \( \cos A \cos B – \sin A \sin B \) |
| \( \cos(A – B) \) | \( \cos A \cos B + \sin A \sin B \) |
| \( \tan(A + B) \) | \( \frac{\tan A + \tan B}{1 – \tan A \tan B} \) |
| \( \tan(A – B) \) | \( \frac{\tan A – \tan B}{1 + \tan A \tan B} \) |
Double Angle Formulas
Set B = A in the compound angle formulas to get double angle formulas. The three forms of cos 2A are particularly important — each form is useful in different integration problems (Class 12, JEE).
| Formula | Expression |
|---|
| \( \sin 2A \) | \( 2\sin A \cos A \) |
| \( \cos 2A \) (form 1) | \( \cos^2 A – \sin^2 A \) |
| \( \cos 2A \) (form 2) | \( 2\cos^2 A – 1 \) |
| \( \cos 2A \) (form 3) | \( 1 – 2\sin^2 A \) |
| \( \tan 2A \) | \( \frac{2\tan A}{1 – \tan^2 A} \) |
Triple Angle Formulas
| Formula | Expression |
|---|
| \( \sin 3A \) | \( 3\sin A – 4\sin^3 A \) |
| \( \cos 3A \) | \( 4\cos^3 A – 3\cos A \) |
| \( \tan 3A \) | \( \frac{3\tan A – \tan^3 A}{1 – 3\tan^2 A} \) |
Half Angle Formulas
| Formula | Expression |
|---|
| \( \sin\frac{A}{2} \) | \( \pm\sqrt{\frac{1 – \cos A}{2}} \) |
| \( \cos\frac{A}{2} \) | \( \pm\sqrt{\frac{1 + \cos A}{2}} \) |
| \( \tan\frac{A}{2} \) | \( \pm\sqrt{\frac{1 – \cos A}{1 + \cos A}} = \frac{\sin A}{1 + \cos A} \) |
Product-to-Sum and Sum-to-Product Formulas
| Type | Formula |
|---|
| Product → Sum | \( 2\sin A \cos B = \sin(A+B) + \sin(A-B) \) |
| Product → Sum | \( 2\cos A \sin B = \sin(A+B) – \sin(A-B) \) |
| Product → Sum | \( 2\cos A \cos B = \cos(A-B) + \cos(A+B) \) |
| Product → Sum | \( 2\sin A \sin B = \cos(A-B) – \cos(A+B) \) |
| Sum → Product | \( \sin C + \sin D = 2\sin\frac{C+D}{2}\cos\frac{C-D}{2} \) |
| Sum → Product | \( \sin C – \sin D = 2\cos\frac{C+D}{2}\sin\frac{C-D}{2} \) |
| Sum → Product | \( \cos C + \cos D = 2\cos\frac{C+D}{2}\cos\frac{C-D}{2} \) |
| Sum → Product | \( \cos C – \cos D = -2\sin\frac{C+D}{2}\sin\frac{C-D}{2} \) |
Inverse Trigonometry Formulas (NCERT Class 12, Chapter 2)
| Function | Domain | Principal Range |
|---|
| \( \sin^{-1} x \) | \( [-1,\,1] \) | \( \left[-\frac{\pi}{2},\,\frac{\pi}{2}\right] \) |
| \( \cos^{-1} x \) | \( [-1,\,1] \) | \( [0,\,\pi] \) |
| \( \tan^{-1} x \) | \( \mathbb{R} \) | \( \left(-\frac{\pi}{2},\,\frac{\pi}{2}\right) \) |
| \( \text{cosec}^{-1} x \) | \( |x| \geq 1 \) | \( \left[-\frac{\pi}{2},\,\frac{\pi}{2}\right] \setminus \{0\} \) |
| \( \sec^{-1} x \) | \( |x| \geq 1 \) | \( [0,\,\pi] \setminus \{\frac{\pi}{2}\} \) |
| \( \cot^{-1} x \) | \( \mathbb{R} \) | \( (0,\,\pi) \) |
| Key Property | Formula |
|---|
| Complementary pair | \( \sin^{-1}x + \cos^{-1}x = \frac{\pi}{2} \) |
| Complementary pair | \( \tan^{-1}x + \cot^{-1}x = \frac{\pi}{2} \) |
| Odd function | \( \sin^{-1}(-x) = -\sin^{-1}x \) |
| Cosine supplement | \( \cos^{-1}(-x) = \pi – \cos^{-1}x \) |
| tan addition formula | \( \tan^{-1}x + \tan^{-1}y = \tan^{-1}\frac{x+y}{1-xy},\; xy<1 \) |
Class-wise NCERT Coverage
| Class | Topics | NCERT Chapter | Exam Relevance |
|---|
| Class 10 | Trig ratios, complementary angle formulas, standard values (0°–90°) | Chapter 8, Chapter 9 | CBSE Board (6–8 marks) |
| Class 11 | Radian/degree, compound angles, multiple angles, all identities above | Chapter 3 | CBSE Board, JEE Main (10–12 marks) |
| Class 12 | Inverse trig functions, properties, equations, integration using trig | Chapter 2, Chapter 7 | CBSE Board, JEE Main/Advanced, NEET |
For the official NCERT Class 10, 11, and 12 Maths textbooks, visit ncert.nic.in. For CBSE sample papers and marking schemes, visit cbse.gov.in.